Series Based on Alternative Theories

In the previous post we established certain series for $1/\pi$ following Ramanujan's technique. These were based on formulas in the classical theory of elliptic functions and integrals. In the field of elliptic functions, Ramanujan surpassed all his predecessors and developed alternative theories which bore striking resemblance to the classical theory and thus provided a grand generalization of the theory of elliptic functions.
In this post we will not develop these alternative theories in detail, but rather try to relate them to the classical theory and thereby provide further series for $1/\pi$.

And then Ramanujan gives a list of 14 series for $ 1/\pi$ without any proof.

If we take a look closely at the definitions above we see that we are talking here about the hypergeometric function
$$K_{s} =\,_{2}F_{1}(s, 1 - s; 1; k^{2})$$
for $ s = 1/3, 1/4, 1/6$ so that Ramanujan's $ K_{1}, K_{2}, K_{3}$ correspond to $ s = 1/4, 1/3, 1/6$ respectively. Also the value of the nome $ q$ is given as
$$q_{s} = \exp\left(-\frac{\pi}{\sin(\pi s)}\frac{K'_{s}}{K_{s}}\right)$$
The traditional theory which is the simplest case corresponds to $ s = 1/2$.

Some further results on these alternative theories were found in Ramanujan's Notebooks and using these Bruce C. Berndt and others developed these theories in detail. We will not discuss any further about these theories. However we will focus on the series for $ 1/\pi$ given by Ramanujan on the basis of these theories.

The last series offered by him is
$$\boxed{\displaystyle \frac{1}{2\pi\sqrt{2}} = \frac{1103}{99^{2}} + \frac{27493}{99^{6}}\frac{1}{2}\frac{1\cdot 3}{4^{2}} + \frac{53883}{99^{10}}\frac{1\cdot 3}{2\cdot 4}\frac{1\cdot 3\cdot 5\cdot 7}{4^{2}\cdot 8^{2}} + \cdots}\tag{1}$$
By looking at the series it is clear that it belongs to the theory based on $ s = 1/4$. The series is very fast converging and each term roughly gives 8 decimal digits. The first term for example gives the approximation
$$\pi \approx \frac{9801}{1103\sqrt{8}} = 3.14159273\ldots$$
correct to 6 places of decimals.

Development by Borwein Brothers

This series was used by Bill Gosper in 1985 to compute the value of $ \pi$ to about 17 million digits. Only problem with this calculation was that there was no proof that the series was correct. A comparison with the value of $ \pi$ obtained from other sources gave a clear indication that the series had to be correct. The series was finally proved shortly after in 1987 by Borwein brothers. Borwein brothers did not develop the alternative theories in detail, but rather showed how these alternative theories could be obtained from the classical theory by suitable transformation of the hypergeometric series involved. Thus they were able to choose different functions $ c(k)$ to express $ (2K/\pi)^{2}$ in various ways as series of powers of $ c(k)$.

Using equations $ (6)$ and $ (7)$ in equations $ (3)$ and $ (4)$ (with $ s = 1/4$) we get
$$\boxed{\displaystyle \frac{2K}{\pi} = (1 + k^{2})^{-1/2}\,_{2}F_{1}\left(\frac{1}{8}, \frac{3}{8}; 1; \left(\frac{g^{12} + g^{-12}}{2}\right)^{-2}\right)}$$
and
$$\boxed{\displaystyle \left(\frac{2K}{\pi}\right)^{2} = (1 + k^{2})^{-1}\,_{3}F_{2}\left(\frac{1}{4},\frac{3}{4},\frac{1}{2}; 1, 1; \left(\frac{g^{12} + g^{-12}}{2}\right)^{-2}\right)}\tag{8}$$
The range of values of $ k$ for which the above formulas are valid is $ 0 \leq k \leq \sqrt{2} - 1$. This limitation comes from the fact that in the formulas $ (3)$ and $ (4)$ we must have $ k \leq k'$ and accordingly we must have $ h \leq h'$. We now have an expression for $ (2K/\pi)^{2}$ in the form desired and we can apply Ramanujan's technique to find a series for $ 1/\pi$. After some reasonable amount of calculation the general formula can be written as:
$$\boxed{\displaystyle \frac{1}{\pi} = \sum_{m = 0}^{\infty}\frac{(1/4)_{m}(3/4)_{m}(1/2)_{m}}{(m!)^{3}}(a + mb)x_{n}^{2m + 1}}\tag{9}$$
where
\begin{align}x_{n} &= \frac{4kk'^{2}}{(1 + k^{2})^{2}} = \frac{2}{g_{n}^{12} + g_{n}^{-12}}\notag\\
(c)_{m} &= c(c + 1)(c + 2)\cdots (c + m - 1)\notag\\
a &= \frac{\sqrt{n}(2g_{n}^{12} - g_{n}^{-12})}{12} - \frac{R_{n}(k, k')}{12}\cdot\frac{g_{n}^{12} + g_{n}^{-12}}{1 + k^{2}}\notag\\
b &= \frac{\sqrt{n}(g_{n}^{12} - g_{n}^{-12})}{2}\notag\end{align}
If $ n = 2$ then $ k = \sqrt{2} - 1$ so that this is the boundary case and it turns out that both $ a$ and $ b$ are zero in this case. Hence in the above general formula we must have $ n > 2$.

Note that the Chudnovsky series given above has not been calculated by directly putting $n = 163$ in the formula $(21)$ (the main difficulty being the calculation of $R_{n}(k, k')$ for $n = 163$), but rather the formula $(21)$ is derived using a different approach through which it is possible to show that the ratio $A/B$ is rational for $n = 163$ and then a numerical calculation of $A/B$ is done and using computer algorithms we find a rational number which matches the numerical value of $A/B$. This alternative approach has been described in a paper by Dr. Bruce C. Berndt and Dr. H. H. Chan.